Final answer:
The values for which the rational expression -3z+6/6z^2+14z-80 is undefined are z = 4 and z = -10/3, found by setting the factors of the denominator to zero.
Step-by-step explanation:
To determine the value(s) for which the rational expression -3z+6/6z^2+14z-80 is undefined, we must find the values of z that make the denominator equal to zero. This is equivalent to solving the quadratic equation 6z^2+14z-80 = 0.
First, we factor the quadratic expression:
- 6z^2+14z-80 can be factored into (2z-8)(3z+10).
Setting each factor equal to zero gives us the values of z that are undefined for the original expression:
- 2z - 8 = 0 implies z = 4
- 3z + 10 = 0 implies z = -10/3
Thus, the rational expression is undefined for z = 4 and z = -10/3.